Express the following sum in closed form. Your answer should be in terms of n.
1. ∑ k=1. 3(k/n) = ?
2. ∑ k=1. (4+2k)^2 = ?
3. ∑ k=1. (3+3* k/n)^2= ?
1 Answer

1.
∑ k=1. 3(k/n)
= (3/n)∑ k=1. (k)
= (3/n)(n/2)(n  1)
= (3/2)(n  1)
2.
∑ k=1. (4+2k)^2
= ∑ k=1. (16 + 16k + 4k^2)
= 16n + 16(n/2)(n  1) + 4[(1/3)n^3 + (1/2)n^2 + (1/6)n]
= 16n + (8n^2  8n) + [(4/3)n^3 + 2n^2 + (2/3)n]
= (4/3)n^3 + 10n^2 + (26/3)n
3.
∑ k=1. (3+3*k/n)^2
= ∑ k=1. (9 + 18*k/n + 9k^2/n^2)
= 9n + (18/n)(n/2)(n  1) + (9/n^2)[(1/3)n^3 + (1/2)n^2 + (1/6)n]
= 9n + (9n  9) + [3n + (9/2) + (3/2n)]
= 21n  9/2 + 3/(2n)